Eisenstein Integers, need help with a proof

Hey all, I am having a problem with an exercise from the book "A Classical Introduction to Modern Number Theory". It involves the ring , where
The question is
For any , show that is congruent to 1, -1 or 0 mod 1 -

I figured the best thing to use was Euclidean division, so I set
and worked out that

I can see how it would work, and have tested it a few times on random Eisenstein integers, but cannot work out how to actually prove it!

Hey all, I am having a problem with an exercise from the book "A Classical Introduction to Modern Number Theory". It involves the ring , where
The question is
For any , show that is congruent to 1, -1 or 0 mod 1 -

I figured the best thing to use was Euclidean division, so I set
and worked out that

I can see how it would work, and have tested it a few times on random Eisenstein integers, but cannot work out how to actually prove it!